a classification for 2-isometries of noncommutative...

A CLASSIFICATION FOR 2-ISOMETRIES OFNONCOMMUTATIVE Lp-SPACES

MARIUS JUNGE∗, ZHONG-JIN RUAN∗, AND DAVID SHERMAN

Abstract. In this paper we extend previous results of Banach, Lamperti

and Yeadon on isometries of Lp-spaces to the non-tracial case first intro-

duced by Haagerup. Specifically, we use operator space techniques and anextrapolation argument to prove that every 2-isometry T : Lp(M)→ Lp(N )

between arbitrary noncommutative Lp-spaces can always be written in theform

T (ϕ1p ) = w(ϕ ◦ π−1 ◦ E)

1p , ϕ ∈ M+

∗ .

Here π is a normal *-isomorphism from M onto the von Neumann subalge-

bra π(M) of N , w is a partial isometry in N , and E is a normal conditionalexpectation from N onto π(M). As a consequence of this, any 2-isometryis automatically a complete isometry and has completely contractively com-plemented range.

1. Introduction

The investigation of isometries on Lp-spaces has a long tradition in the theoryof Banach spaces and has connections to probability and ergodic theory. Banach[2] considered the discrete case and showed that for a surjective isometry on `p

one deduces the existence of a permutation π: N → N and a sequence of scalars{λn} with unit modulus so that T (en) = λneπ(n). Banach also stated, withoutproof, the form of surjective isometries of Lp([0, 1],m). These are weighted

composition operators; i.e.

(1.1) T (f)(x) = h(x)f(ϕ(x)), x ∈ [0, 1],

where ϕ is a measurable bijection of the unit interval and h is a measurablefunction with |h|p = d(m◦ϕ)

dm . As shown by Lamperti [24], this paradigm is stillbasically correct for non-surjective isometries on general Lp(X,Σ, µ), but the

Date: March 5, 2004.1991 Mathematics Subject Classification. Primary 46L07 and 46L05; Secondary 47L25.Key words and phrases. von Neumann algebra, noncommutative Lp space, isometry, oper-

ator space.∗ The first and second authors were partially supported by the National Science Foundation

DMS-0088928 and DMS-0140067.

1

2 MARIUS JUNGE∗, ZHONG-JIN RUAN∗, AND DAVID SHERMAN

“composition” is defined in terms of a mapping on the measurable sets (a so-called regular set isomorphism, see [24] or [7]). For a sufficiently nice measurespace, there is still an underlying point mapping (see [12]). Notice that a set map-ping is nothing but a map on the projections in the associated L∞-algebras, soLamperti’s result can be formulated naturally in terms of von Neumann algebras!Moreover, this formulation implies that isometries must preserve disjointness

of support, which has been a key step in every stage of the classification ofLp-isometries, including the present paper.

The most familiar noncommutative Lp-space is the Schatten-von Neumann p-ideal Sp [26], the class of operators in K(H) (or in B(H)) whose singular values arep-summable. Segal [37] extended this concept to general semifinite von Neumannalgebras. In this category, Lp-isometries have been investigated by Broise [4],Russo [36], Arazy [1], Katavolos [17], [18], [19], and Tam [46]. In 1981, Yeadon[53] finally gave a satisfactory answer.

Theorem 1. [53, Theorem 2] Let M and N be two semifinite von Neumannalgebras with traces τM and τN , respectively. For 1 ≤ p <∞ and p 6= 2, a linearmap

T : Lp(M, τM) → Lp(N , τN )

is an isometry if and only if there exist a normal Jordan *-monomorphismJ : M → N , a partial isometry w ∈ N , and a positive self-adjoint operatorB affiliated with N such that the spectral projections of B commute with J(M),which verify the following conditions:

(1) w∗w = J(1) = s(B);(2) τM(x) = τN (BpJ(x)) for all x ∈M+;(3) T (x) = wBJ(x) for all x ∈M∩ Lp(M, τM).

Moreover, J , w, and B are uniquely determined by the above conditions (1)−(3).

At approximately the same time, Haagerup [9] discovered a way to constructnoncommutative Lp-spaces from arbitrary von Neumann algebras. While manypeople worked on general noncommutative Lp-spaces over the last twenty years,until recently little progress had been made on the classification of their isome-tries. Watanabe wrote a series of papers (including [49] and [50]) supplying

Lp 2-ISOMETRIES 3

many ideas and some partial results. In [38], Watanabe’s techniques are devel-oped to obtain classification results for Lp-isometries which are complementaryto those established in the present paper (by entirely different methods), and in[39] a canonical form of surjective Lp-isometries is derived, proving that Lp(M)and Lp(N ) are isometrically isomorphic if and only if M and N are Jordan*-isomorphic.

In this paper, we use operator space techniques to give a complete classifica-tion of 2-isometries on general noncommutative Lp-spaces. We begin by extend-ing Yeadon’s result to isometries between noncommutative Lp-spaces where onlythe initial von Neumann algebra is assumed semifinite (using a recent result ofRaynaud/Xu [34]). After some background information (section 2) this task isperformed in section 3. If we drop the assumption that the initial algebra issemifinite, then Lp(M) ∩M = {0}, and a new kind of problem occurs. There isno longer a canonical embedding of the von Neumann algebra (or finite elementstherein) into the noncommutative Lp-space, and therefore we may no longeruse the lattice of projections canonically. This has been the main obstructionfor applying Yeadon’s groundbreaking technique [53]. We found that operatorspace methods can be employed to overcome this difficulty. The operator spacestructure of noncommutative Lp-spaces was introduced by Pisier [30] and laterextended to the non-semifinite case (see [14] and [31], the structure in [6] is notentirely compatible here). In section 4 we are essentially interested in takingcolumns and rows of elements in Lp in order to classify complete isometries.Previous work for the case p = 1 was done by [28] and [27]. While isometriesare connected with Jordan structure, a 2-isometry must already preserve multi-plicative structure. This is one of the key observations for our result, the otherbeing an extrapolation argument which shows that if we have an Lp-isometry forone p 6= 2, then we have an associated isometry for any other p. The case p = 4connects with the theory of selfpolar forms (from [10]) and thus can be used toconstruct a conditional expectation, leading to a proof of the main theorems.The final section of the paper consists of remarks.

To be more specific let us fix some notation. If N is a von Neumann algebra,then Lp(N ) is the linear span of elements ϕ

1p , where ϕ ranges over the positive

states in N∗.


Theorem 2. Let 1 ≤ p 6= 2 <∞ and T : Lp(M) → Lp(N ) be a linear map. Thefollowing are equivalent:

i) id⊗ T : Lp(M2 ⊗M) → Lp(M2 ⊗N ) is an isometry,ii) id⊗ T : Lp(Mm ⊗M) → Lp(Mm ⊗N ) is an isometry for all m ∈ N,iii) There exists an injective normal *-homomorphism π: M→N , a partial

isometry w ∈ N , and a conditional expectation E from N onto π(M)such that

(1.2) T (ϕ1p ) = w(ϕ ◦ π−1 ◦ E)

1p , ϕ ∈M+

∗ .

In (1.2) we have to use the fact that a map defined on positive vectors extendsuniquely to the whole of Lp(N ). Moreover, we recall that Lp(N ) is an N − Nbimodule, so that wϕ

1p is a well-defined element satisfying ‖wϕ

1p ‖p = ϕ(1)

1p (see

[47] for more details). In the σ-finite case, φ1pN is dense in Lp(N ) for every

normal faithful state φ. Then we may alternatively describe T by

(1.3) T (φ1px) = w(φ ◦ π−1 ◦ E)

1pπ(x), x ∈M.

Notice that when φ is not tracial, the inclusion mapping ip: N → Lp(N ), ip(x) =φ

1px, no longer preserves disjoint left supports. This is the obstacle mentioned

above.

To end this introduction let us mention some recent sources which will providefurther background to interested readers. The book of Fleming and Jamison [7]is an up-to-date survey on isometries in Banach spaces (including classical Lp-spaces); their companion volume will treat the case of noncommutative Lp-spaces.The handbook article of Pisier and Xu [32] provides a general overview of non-commutative Lp-spaces, focusing on Banach space properties and including a richbibliography. There is also recent groundbreaking work on the non-isometric

isomorphism/embedding question by Haagerup, Rosenthal, and Sukochev [11]which is entirely disjoint to the isometric analysis of this paper.

2. Some background

In this section we provide some background on noncommutative Lp-spaces andtheir operator space structure. The readers are referred to Nelson [25], Kosaki[22], Terp [47] and [48] for details on noncommutative Lp-spaces and to Pisier[30], [31], Fidaleo [6], and Junge-Ruan-Xu [14] for the canonical operator space

Lp 2-ISOMETRIES 5

structure on these spaces. Pisier and Xu’s recent survey paper [32] provides anice overview of this subject. For general background on the modular theory ofvon Neumann algebras, we recommend the Takesaki oeuvre [43], [44] and [45],and Stratila [41].

We first recall the Lp-space (1 ≤ p < ∞) associated with a semifinite algebraM equipped with a given normal faithful semifinite trace τ (simply called a“trace” from here on). Consider the set

{T ∈M | ‖T‖p , τ(|T |p)1/p <∞}.

It can be shown that ‖ · ‖p defines a norm on this set. Then the norm comple-tion, which is denoted Lp(M, τ), is the noncommutative Lp-space obtained from(M, τ). It turns out that one can identify elements of Lp(M, τ) with certainτ -measurable operators affiliated with M (see [25]). Clearly τ is playing the roleof integration here.

A von Neumann algebra lacking a faithful trace is not amenable to the previousdefinition. There are several alternative constructions which work in full gener-ality. Let us first recall the construction initiated by Haagerup [9] and carriedout in detail by Terp [47]. Choose a normal faithful semifinite weight φ on M.We consider the one-parameter modular automorphism σφt (associated with φ)on M and obtain a semifinite von Neumann algebra M , M oσφ R which hasan induced trace τ and a trace-scaling dual action θ such that τ ◦ θs = e−sτ forall s ∈ R.

The original von Neumann algebra M can be identified with a θ-invariantvon Neumann subalgebra L∞(M) of M. For 1 ≤ p < ∞, the noncommutativeLp-space Lp(M, φ) is defined to be the space of all (unbounded) τ -measurableoperators affiliated with M such that θs(T ) = e−

spT for all s ∈ R. It is known

from Terp [47, Chapter II] that there is a one-to-one correspondence betweenbounded (positive) linear functionals ψ ∈ M∗ and τ -measurable (positive self-adjoint) operators hψ ∈ L1(M, φ) under the connection given by

ψ(x) = τ(hψx), x ∈ M,

where ψ is the so-called dual weight for ψ. This correspondence actually extendsto all of M∗ and L1(M, φ), and we may define the “tracial” linear functional


tr = trM: L1(M, φ) → C by(2.1)tr(hψ) = ψ(1), satisfying tr(hψx) = tr(xhψ) = ψ(x), ψ ∈M+

∗ , x ∈M.

Given any h ∈ Lp(M), we have the polar decomposition h = w|h|, where |h|is a positive operator in Lp(M)+ and w is a partial isometry contained in Msuch that the projection sl(h) = ww∗ is the left support of h and the projectionsr(h) = w∗w is the right support of h. (We simply use s for the support ofpositive vectors, operators, and maps.) We can define a Banach space norm onLp(M, φ) by

(2.2) ‖h‖p = tr(|h|p)1p = ψ(1)

1p

if ψ ∈ M∗ corresponds to |h|p ∈ L1(M, φ)+. With this norm, it is easy to seethat L1(M, φ) is isometrically and orderly isomorphic to M∗. We note that upto isometry the noncommutative Lp-space constructed above is actually indepen-dent of the choice of normal faithful semifinite weight on M. Therefore, we willsimply write Lp(M) if there is no confusion.

We note that for any positive operator h ∈ Lp(M)+, hp is a positive operatorin L1(M)+ and thus we can write hp = hψ for a corresponding positive linearfunctional ψ ∈ M+

∗ . Therefore, we may identify h with ψ1p , i.e. we can simply

write

h = ψ1p .

This notation, discussed specifically in [52], [5, Section V.B.α] and [40], providesthe relations

ψitϕ−it = (Dψ : Dϕ)t, ϕitxϕ−it = σϕt (x), ϕ, ψ ∈M+∗ , ϕ faithful.

It will be used in section 4, where it suggests that the main results really dealwith “noncommutative weighted composition operators”.

For 1 ≤ p <∞, we let Lp(M)′ denote the dual space of Lp(M). Then we canobtain the isometric isomorphism Lp(M)′ = Lp′(M) under the trace duality

(2.3) 〈x, y〉 = tr(xy) = tr(yx)

for all x ∈ Lp(M) and y ∈ Lp′(M). (Throughout p′ denotes the conjugateexponent of p; i.e. 1

p + 1p′ = 1.)

One of the advantages in using Haagerup’s approach is that the naturalM−Mbimodule structure on Lp(M) is just operator composition. It was shown in [15,

Lp 2-ISOMETRIES 7

Lemma 1.2] that for any h ∈ Lp(M), we have

(2.4) {xh | x ∈M} = Lp(M)sr(h) and {hx | x ∈M} = sl(h)Lp(M).

In particular, if M is a σ-finite von Neumann algebra and φ is a normal faithfulpositive linear functional on M, then h = φ

1p is a cyclic vector in Lp(M), i.e.

{xφ1p | x ∈M} and {φ

1px | x ∈M}

are norm dense in Lp(M). In this case, we can obtain the following result (seeJunge and Sherman [15, Lemma 1.3]).

Lemma 2.1. Let φ ∈ M+∗ be faithful. A bounded net {xα} ∈ M converges

strongly to x, xαs→ x, if and only if xαφ

1p → xφ

1p (or φ

1pxα → φ

1px ) in Lp(M).

Next let us recall Kosaki’s complex interpolation construction of Lp(M) (see[22]). We assume that M is a σ-finite von Neumann algebra and φ is a normalfaithful state on M. Then we may identify M with a subspace of M∗ by theright embedding

x ∈M 7→ φx ∈M∗,

where 〈φx, y〉 = φ(xy). We use the right embedding (instead of the left embed-ding) in this paper for the convenience of our notation in representation theorems.Then the complex interpolation [M,M∗] 1

pis a Banach space which can be iso-

metrically identified with φ1p′ Lp(M) by∥∥∥φ 1

p′ h∥∥∥

[M,M∗] 1p

= ‖h‖Lp(M)

for all h ∈ Lp(M) (see [22, Theorem 9.1]). In particular, any φ1px ∈ Lp(M)

corresponds to an element φx in [M,M∗] 1p

since

(2.5) φx = φ1p′ (φ

1px).

In this case, we have

(2.6) ‖φx‖[M,M∗] 1p

=∥∥∥φ 1

px∥∥∥Lp(M)

.

We will simply write Lp(M) = [M,M∗] 1p

when there is no confusion.Using the complex interpolation, Pisier [30] contructed a canonical operator

space matrix norm

(2.7) Mn(Lp(M)) = [Mn(M),Mn(Mop∗ )] 1

p


on Lp(M) (also see [31] and [14]). For each n ∈ N, we may also consider thenoncommutative Snp -integral Snp [Lp(M)] of Lp(M) and it turns out that we havethe isometric isomorphism

(2.8) Snp [Lp(M)] = Lp(Mn⊗M).

Moreover we can recover the canonical matrix norm on Lp(M) by

(2.9) ‖x‖Mn(Lp(M)) = sup{‖αxβ‖Snp [Lp(M)] : ‖α‖Sn

2p, ‖β‖Sn

2p≤ 1}.

Consequently, a linear map T : Lp(M) → Lp(N ) is a complete contraction (re-spectively, a complete isometry) if and only if for every n ∈ N,

idSnp⊗ T : Snp [Lp(M)] → Snp [Lp(N )]

is a contraction (respectively, an isometry).Finally let us recall the following equality condition for the noncommutative

Clarkson inequality, which was shown for semifinite von Neumann algebras byYeadon [53], for general von Neumann algebras with 2 < p <∞ by Kosaki [21],and just recently for all p 6= 2 by Raynaud and Xu [34].

Theorem 2.2. For h, k ∈ Lp(M), 0 < p 6= 2 <∞,

(2.10) ‖h+ k‖pp + ‖h− k‖pp = 2(‖h‖pp + ‖k‖pp) ⇐⇒ hk∗ = h∗k = 0.

Let us agree to say that Lp vectors h and k are orthogonal when they satisfythe conditions in (2.10). We mentioned earlier that isometries on classical Lp-spaces preserve disjointness of support. Theorem 2.2 tells us that isometries ongeneral noncommutative Lp-spaces preserve orthogonality. This plays a key rolein several of our proofs.

3. A generalization of Yeadon’s theorem

In 1981, Yeadon obtained a very satisfactory and complete description forisometries between noncommutative Lp-spaces associated with semifinite vonNeumann algebras (Theorem 1 above). We found that an analog of Yeadon’sresult still holds if the initial algebra M is semifinite and the range algebra N isarbitrary. The proof of this generalized result is almost the same as that given inYeadon [53]. The major difference is that if N is a general von Neumann algebrawith a normal faithful semifinite weight φ, the positive self-adjoint operator B is

Lp 2-ISOMETRIES 9

affiliated with the semifinite von Neumann algebra crossed product N = Noσφ R(instead of N ), and condition (2) in Theorem 1 should be revised as

τM(x) = trN (BpJ(x))

for all x ∈ M+, where trN is the Haagerup trace introduced in (2.1). We willoutline the proof of this generalized result in the following theorem since it willprovide us necessary notations and motivation for the rest of the paper. We notethat a proof by different methods can be found in Sherman [38]. Yeadon couldnot have proven such a result because general Lp-spaces were only being inventedas he was writing his paper, but he did prove it for preduals (the p = 1 case) in[53, §4].

In the rest of this section, let us assume that M is a semifinite von Neumannalgebra with a (normal faithful semifinite) trace τM and N is an arbitrary vonNeumann algebra with a normal faithful semifinite weight φ, which induces thenormal faithful semifnite trace τN on the crossed product N = N oσφ R, andthe Haagerup trace trN on L1(N ).

Theorem 3.1. For 1 ≤ p <∞ and p 6= 2, a linear map

T : Lp(M, τM) → Lp(N )

is an isometry if and only if there exist a normal Jordan *-monomorphismJ : M→N , a partial isometry w ∈ N , and a positive self-adjoint operator B af-filiated with N such that θs(B) = e−

spB for all s ∈ R and the spectral projections

of B commute with J(M) ⊂ N ⊂ N , which verify the following conditions:

(1) w∗w = J(1) = s(B);(2) τM(x) = trN (BpJ(x)) for all x ∈M∩ Lp(M);(3) T (x) = wBJ(x) for all x ∈M∩ Lp(M, τM).

Moreover, J , w, and B are uniquely determined by the above conditions (1)−(3).

Proof. First note that the stated conditions do define an isometry; (2) implies

‖x‖pp = τM(|x|p) = trN (BpJ(|x|p)) = trN (|BJ(x)|p) = ‖T (x)‖pp

for any x ∈ M⋂Lp(M). In the rest of the proof we derive these conditions for

an arbitrary isometry T .For each τM-finite projection e ∈M, we let T (e) = weBe be the polar decom-

position of T (e) ∈ Lp(N ). Then we is a partial isometry in N and Be = |T (e)|


is a positive element of Lp(N ) such that

(3.1) Bew∗ewe = Be = w∗eweBe.

If we define J(e) = w∗ewe = sr(T (e)) = s(Be) to be the corresponding projectionin N , then Be commutes with J(e).

Let e and f be two mutually orthogonal τM-finite projections in M. Since Tis an isometry, we have

‖T (e)± T (f)‖pp = ‖T (e± f)‖pp = ‖e± f‖pp = ‖e‖pp + ‖f‖pp = ‖T (e)‖pp + ‖T (f)‖pp.

Then the Clarkson inequality is an equality, and we can conclude by Theorem2.2 that

T (e)∗T (f) = T (e)T (f)∗ = 0.

This implies that

BeBf = 0, J(e)J(f) = 0, and w∗ewf = wew∗f = 0.

The linearity of T gives we+fBe+f = weBe + wfBf , from which

B2e+f = (we+fBe+f )∗(we+fBe+f ) = (weBe + wfBf )∗(weBe + wfBf )

= B2e +B2

f = (Be +Bf )2

and so

weBe + wfBf = we+fBe+f = we+fBe + we+fBf .

This implies

(3.2) Be+f = Be +Bf , J(e+ f) = J(e) + J(f), and we+f = we + wf .

If x =∑λiei ∈ M is a self-adjoint simple operator with mutually orthogonal

τ -finite projections {ei} in M, we define J(x) =∑λiJ(ei). It is easy to verify

that

J(x2) = J(x)2 and ‖J(x)‖∞ = ‖x‖∞.

Moreover, we have J(λx) = λx for all real λ and J(x + y) = J(x) + J(y) ifx and y are commuting self-adjoint simple operators in M. In general, for anyself-adjoint operator x ∈M, there exists a sequence of simple functions fn on thespectrum of x with fn(0) = 0 which converges uniformly to the identity functionf(λ) = λ. We define J(x) to be the ‖ · ‖∞-limit of J(fn(x)) in N .

Now fix a τM-finite projection e ∈ M. If f is a projection in M such thatf ≤ e, then T (f) = T (e)J(f) and thus

(3.3) T (x) = T (e)J(x) = weBeJ(x)

Lp 2-ISOMETRIES 11

for all x ∈ eMsae. It follows that for any x, y ∈ eMsae,

T (e)(J(x+ y)− J(x)− J(y)) = T (x+ y)− T (x)− T (y) = 0.

This implies

J(x+ y)− J(x)− J(y) = 0

and thus J is a real linear map from eMsae into Nsa. Next we can extend J toa *-linear map on all of eMe by

J(x+ iy) = J(x) + iJ(y), ∀x, y ∈ eMsae.

Still on eMe, we have

J [(x+ iy)2] = J [x2 − y2 + i((x+ y)2 − x2 − y2)]

= J(x)2 − J(y)2 + i((J(x) + J(y))2 − J(x)2 − J(y)2)

= [J(x+ iy)]2.

This shows that J is a Jordan *-monomorphism on eMe. The normality of J fol-lows from (3.3) and Lemma 2.1. It is clear from (3.1) and the above constructionthat all spectral projections of Be commute with J(eMe).

If τM is a finite trace, we may take B = B1 and the theorem is proved. If τMis not finite, some gluing must be done. In this case, we may assume that {eα} isthe (increasing) net of all τM-finite projections in M. Then eα → 1 in the strongoperator topology. By (3.2) and (3.3), we may find a partial isometry w ∈ N (asa strong limit of {weα

}) and a positive self-adjoint operator B (as the supremumof {Beα

}) affiliated with N , and extend J to a normal Jordan *-monomorphismfrom M into N such that conditions (1) − (3) are satisfied. In this case, thespectral projections of B commute with J(M), and B satisfies θs(B) = e−

spB

for all s ∈ R. We also have

(3.4) weα= wJ(eα), Beα

= BJ(eα),

and

(3.5) trN (Bpy) = sup{trN (Bpeαy)}

for every y ∈ N+. Since each Bpeα∈ L1(N ) corresponds to a normal positive

linear functional ϕα = trN (Bpeα·) on N , Bp corresponds to a normal semifinite

weight ϕ = trN (Bp·) onN , which has support J(1). Therefore, ϕ is faithful whenrestricted to J(1)NJ(1). The uniqueness of J,w, and B is an easy consequencewhich we leave to the reader. �


If we assume J(1) = 1 then ϕ is a normal faithful semifinite weight on N . Inthis case, we may write Lp(N ) = Lp(N , ϕ) and it follows from the conditions (2)and (3) in Theorem 3.2 that the Jordan *-monomorphism J : M → N inducesan orderly isometric injection Jϕp from Lp(M, τM) into Lp(N , ϕ), which is givenby

(3.6) Jϕp (x) = BJ(x)

for all x ∈ M⋂Lp(M, τ). If J(1) 6= 1, then we may orderly and isometrically

identify Lp(M, τM) with a subspace of Lp(J(1)NJ(1), ϕ) ' J(1)Lp(N )J(1).Using a standard matricial technique, we obtain the following equivalence re-

sult. The p = 1 case is mentioned (without proof) in [27].

Proposition 3.2. Let M be a semifinite von Neumann algebra and let N be anarbitrary von Neumann algebra. Let 1 ≤ p < ∞ and p 6= 2. For an isometryT = wBJ : Lp(M, τM) → Lp(N ), the following statements are equivalent:

(i) T is a complete isometry,(ii) T is a 2-isometry,(iii) the Jordan map J : M→N is multiplicative.

Proof. It is obvious that (i) ⇒ (ii).If J is multiplicative from M into N then Jn = idn ⊗ J is a *-isomorphism

from Mn(M) into Mn(N ) for every n ∈ N. In this case, we can write

idSnp⊗ T : Snp [Lp(M)] = Lp(Mn⊗M, trn ⊗ τM) → Snp [Lp(N )] = Lp(Mn⊗N )

as

idSnp⊗ T = wnBnJn

Here wn = In ⊗ w is a partial isometry in Mn(N ) such that w∗nwn = Jn(1n). Ifwe let B = sup{Beα

} as in the proof of Theorem 3.1, then the positive selfadjointoperator Bn = supα(Beα

⊕p Beα⊕p · · · ⊕p Beα

) is affiliated with Mn(N ) whosespectral projections commute with Jn(Mn(M)). We have that

(trn ⊗ τM)(x) = (trn ⊗ trN )(BpnJn(x))

for all x ∈ Mn(M)+. Then each idSnp⊗ T : Snp [Lp(M)] → Snp [Lp(N )] is an

isometry by Theorem 3.1. Therefore, T is a complete isometry and we proved(iii) ⇒ (i).

It remains to prove (ii) implies (iii). First T is an isometry and thus has therepresentation T = wBJ by Theorem 3.1. Since T is also a 2-isometry, we deduce

Lp 2-ISOMETRIES 13

an isometry

T = idS2p⊗ T : S2

p [Lp(M, τM)] → S2p [Lp(N )].

Applying Theorem 3.1 to T , we obtain a normal Jordan *-monomorphism

J : M2⊗M →M2⊗N ,

a partial isometry w ∈ M2⊗N , and a positive selfadjoint operator B satisfyingthe conditions in Theorem 3.1.

If ei (i = 1, 2) are τM-finite projections inM, then e =[e1 00 e2

]is a (tr2⊗τM)-

finite projection inM2⊗M. Let T (e) = weBe be the polar decomposition of T2(e)and let T (ei) = wei

Beibe the polar decomposition of T (ei) (i = 1, 2). Since

T (e) =[T (e1) 0

0 T (e2)

]=[we1 00 we2

] [Be1 00 Be2

]we must have, by the uniqueness of the polar decomposition,

we =[we1 00 we2

]and Be =

[Be1 00 Be2

].

According to the definition of J given in the proof of Theorem 3.1, we have

J

([e1 00 e2

])=[J(e1) 0

0 J(e2)

],

and thus we can conclude that

J

([x 00 y

])=[J(x) 0

0 J(y)

]for all x, y ∈M. Since T is an M2-bimodule morphism, we can write

T

([0 xy 0

])= T

([0 11 0

] [x 00 y

])=[0 11 0

] [T (x) 0

0 T (y)

].

Then for any x, y ∈M, we obtain

J

([0 xy 0

])=[

0 J(x)J(y) 0

].

From this we can conclude that J(xy) = J(x)J(y) since[J(xy) 0

0 J(yx)

]= J

([xy 00 yx

])= J

([0 xy 0

]2)= J

([0 xy 0

])2

=[

0 J(x)J(y) 0

] [0 J(x)

J(y) 0

]=[J(x)J(y) 0

0 J(y)J(x)

].

Therefore J is multiplicative. Finally we note that we can conclude from theabove calculations that J = J2, w = w2 and B = B2. �


If T = wBJ : Lp(M, τM) → Lp(N ) is a 2-isometry (or equivalently, a completeisometry) with J a normal *-monomorphism from M into N , then

(3.7) S = w∗T = BJ

is a completely positive and completely isometric injection from Lp(M, τM) intoLp(N ). Therefore, we can completely orderly and completely isometrically iden-tify Lp(M, τM) with the operator subspace S(Lp(M, τM)) of Lp(N ).

Proposition 3.3. Let T : Lp(M, τM) → Lp(N ) be a 2-isometry (or equivalently,a complete isometry). Then T (Lp(M, τM)) is completely contractively comple-mented in Lp(N ).

If, in addition, T is positive, then T (Lp(M, τM)) is completely positively andcompletely contractively complemented in Lp(N ).

Proof. Assume the notation of Theorem 3.1. We have that J is multiplicative byProposition 3.2, so that J(M) is a von Neumann subalgebra of N . Without lossof generality, we may assume that ϕ = trN (Bp·) is faithful on N . Otherwise werestrict our argument to J(1)NJ(1).

Since Bp is the Pedersen-Takesaki derivative [29] of ϕ on N with respect toτN , we have that

σϕt (y) = σϕt (y) = BiptyB−ipt, y ∈ N ⊂ N .

In particular, we have

σϕt (J(x)) = BiptJ(x)B−ipt = J(x)

for all x ∈ M, since B commutes with J(M). Then we can conclude fromTakesaki’s theorem [42] that there exists a unique normal conditional expectationE from N onto J(M) such that

ϕ ◦ E = ϕ.

From this, we may induce a completely positive and completely contractive pro-jection Ep from Lp(N ) = Lp(N , ϕ) onto w∗T (Lp(M, τ)) (see [11], [16], [38], orthe construction sketched after Lemma 4.8). Then h 7→ wEp(w∗h) is the requiredprojection. If T is positive, w and w∗ may be omitted, so this projection is Epitself. �

Lp 2-ISOMETRIES 15

Proposition 3.3 is a noncommutative version of the classical fact that for anyisometric embedding

T : Lp(X,ΣX , µX) → Lp(Y,ΣY , µY ),

the image space T (Lp(X,ΣX , µX)) must be contractively complemented in thespace Lp(Y,ΣY , µY ) (see Lacey [23]). Stronger results are in Theorem 4.9 of thispaper and Section 8 of [38].

4. 2-isometries on general noncommutative Lp-spaces

The aim of this section is to study 2-isometries on general noncommutativeLp-spaces. We will use the alternative notation, i.e. positive linear functionals

ϕ1p , instead of positive self-adjoint operators h

1pϕ , for elements in Lp(M)+. Until

the proof of Theorem 2 given at the end of the section, we assume that M is anarbitrary σ-finite von Neumann algebra with a fixed normal faithful state φ. Wealso use φ for its corresponding density operator hφ in L1(M), so that φ

1pM is

norm dense in Lp(M).As in §3, we start by using support projections to construct an embedding

from M into N . But since M⋂Lp(M) = {0} (p 6= ∞) when M is not semifi-

nite, we cannot directly employ the projection lattice of M, and instead workwith the vectors {ϕ

1px | x ∈ M}. Then 2 × 2 matrix equations produce the

*-monomorphism π (in the next proposition), but several steps are still requiredto show that π(M) is complemented and obtain the desired decomposition forT .

Proposition 4.1. Let 1 ≤ p < ∞ and p 6= 2. If T : Lp(M) → Lp(N ) is a2-isometry, then there exists a normal *-monomorphism π: M→N such that

(4.1) T (φ1px) = T (φ

1p )π(x) , x ∈M.

Moreover, π does not depend on the choice of (faithful) φ ∈M+∗ .

Proof. We first define a map π between projection lattices by

(4.2) π: e ∈ P(M) 7→ sr(T (φ1p e)) ∈ P(N ).

That is, with the polar decomposition T (φ1p e) = we|T (φ

1p e)|, we set π(e) = w∗ewe.

If e ⊥ f , then [φ

1p e 00 0

]and

[0 0

φ1p f 0

]


are two orthogonal elements in S2p [Lp(M)]. By Theorem 2.2, their images[

T (φ1p e) 0

0 0

]and

[0 0

T (φ1p f) 0

]under idS2

p⊗ T are orthogonal in S2

p [Lp(N )]. This shows that

sr

([T (φ

1p e) 0

0 0

])=[sr(T (φ

1p e)) 0

0 0

]=[π(e) 0

0 0

]is orthogonal to

sr

([0 0

T (φ1p f) 0

])=[sr(T (φ

1p f)) 0

0 0

]=[π(f) 0

0 0

],

and thus π(e) is orthogonal to π(f). Since

T (φ1p )π(e) = [T (φ

1p e) + T (φ

1p (1− e))]π(e)

= T (φ1p e) + T (φ

1p )π(1− e)π(e) = T (φ

1p e),

we have (4.1) for projections in M. As a consequence, we obtain

π(e+ f) = π(e) + π(f)

for orthogonal projections e ⊥ f since

T (φ1p )π(e+ f) = T (φ

1p (e+ f)) = T (φ

1p e) + T (φ

1p f) = T (φ

1p )(π(e) + π(f))

and T (φ1p ) is separating for the right action of π(1)Nπ(1).

Now we extend π as in the proof of Theorem 3.1: first to finite real linearcombinations of orthogonal projections, then to all self-adjoint elements in M bycontinuity, and finally to all of M by complex linearity. Apparently π satisfies

T (φ1px) = T (φ

1p )π(x), x ∈M.

This relation implies additivity: for x, y ∈M, we have

T (φ1p )π(x+ y) = T (φ

1p (x+ y)) = T (φ

1px) + T (φ

1p y) = T (φ

1p )(π(x) + π(y)).

Now let u be a unitary element of M. Replacing φ1p by φ

1pu in the definition

of π, we obtain a new *-preserving map πu. For a projection e in M,[φ

1p e 00 0

]and

[0 0

φ1pu(1− e) 0

]are orthogonal, so π(e) ⊥ πu(1− e). Since

πu(1) = sr(T (φ1pu)) = sr(π(u)) = π(1),

Lp 2-ISOMETRIES 17

we must have π(e) = πu(e) for every projection e, whence π = πu. Then for anyx ∈M,

T (φ1p )π(ux) = T (φ

1pux) = T (φ

1pu)π(x) = T (φ

1p )π(u)π(x).

Since the linear span of the unitaries is all ofM, it follows that π is multiplicative.Then

T (φ1pxy) = T (φ

1p )π(xy) = T (φ

1p )π(x)π(y) = T (φ

1px)π(y), x, y ∈M,

(densely) establishes (4.1) and the independence of π from the choice of φ.To prove the normality of π, we let {xα} be a bounded net in M such that

xαs→ x in the strong topology. It follows from Lemma 2.1 that φ

1pxα → φ

1px in

Lp(M), and thus

T (φ1p )π(xα) = T (φ

1pxα) → T (φ

1px) = T (φ

1p )π(x)

in Lp(N ). Again by Lemma 2.1, this implies π(xα) s→ π(x) in π(1)Nπ(1) andthus in N . �

Duality will be a very important tool in the following arguments. Given abounded linear map T : Lp(M) → Lp(N ), we let T ′: Lp′(N ) → Lp′(M) denotethe adjoint of T and let T

′∗ = (T ′)∗ denote the *-adjoint of T ′, i.e.,

T′∗(k) = T ′(k∗)∗ k ∈ Lp(N ).

Then

trM(T′∗(k)∗h) = trN (k∗T (h)) , h ∈ Lp(M), k ∈ Lp′(N ),

defines a sesquilinear form on Lp(M) × Lp′(N ). We also set the notation ϕ ,

|T (ϕ1p )|p for any state ϕ ∈M+

∗ . Thus ϕ1p is the absolute value of T (ϕ

1p ).

Lemma 4.2. Let 1 ≤ p < ∞ and let T : Lp(M) → Lp(N ) be a 2-isometry (oran isometry satisfying (4.1)). Then

φ ◦ π = φ .

Proof. Let us first assume that p 6= 1. Since φ is a normal faithful state in M+∗ ,

then φ1p is a unit vector in Lp(M) since

‖φ1p ‖pp = φ(1) = 1.

This implies that T (φ1p ) = wφ

1p is a unit vector in Lp(N ) and thus

1 = trN ((wφ1p′ )∗T (φ

1p )) = trM(T

′∗(wφ1p′ )∗φ

1p ) .


Since Lp′(M) is uniformly convex, φ1p admits exactly one norm attaining element.

Thus we must have

T′∗(wφ

1p′ ) = φ

1p′ .

On the other hand, we have

φ(x) = trM((φ1p′ )∗φ

1px) = trM(T

′∗(wφ1p′ )∗φ

1px) = trN ((wφ

1p′ )∗T (φ

1px))

= trN (φ1p′ w∗wφ

1pπ(x)) = φ(π(x)) .

If p = 1, the same argument applies, replacing φ1p′ and φ

1p′ by 1. (Since φ is

faithful, it attains its norm at 1 only.) �

We can already say quite a lot in case p = 1.

Proposition 4.3. Let T : M∗ → N∗ be a 2-isometry. There are a normal *-monomorphism π: M→N , a partial isometry w ∈ N , and a normal conditionalexpectation E: N → π(M) such that

T (ϕ) = w(ϕ ◦ π−1 ◦ E), ϕ ∈M∗.

Proof. It was shown in [38, Theorem 3.2] that for any such L1-isometry T , thereare a normal Jordan *-monomorphism J : M → N , a partial isometry w ∈ N ,and a normal positive projection P : N → J(M), faithful on J(1)NJ(1), suchthat

T (ϕ) = w(ϕ ◦ J−1 ◦ P ), ϕ ∈M∗.

(We note that an equivalent result was proved earlier by Kirchberg [20].) So theinduced map S = w∗T is a positive isometry and thus we can get ϕ = ϕ◦J−1 ◦Pfor all ϕ ∈ M+

∗ . Precomposing with the π from Proposition 4.1 and applyingLemma 4.2,

ϕ ◦ J−1 ◦ P ◦ π = ϕ ◦ π = ϕ, ∀ faithful ϕ ∈M+∗ .

Apparently J−1◦P ◦π is the identity map. In particular, we have P (π(e)) = J(e)for any projection e in M. Using properties of P and J (see [38, Lemma 5.4]),we can conclude that

P (J(e)π(e)J(e)) = J(e)P (π(e))J(e) = [J(e)]3 = J(e) = P (J(e)).

Since J(e)π(e)J(e) ≤ J(e) and J(1) = s(P ), we must have J(e)π(e)J(e) = J(e).This implies π(e) ≥ J(e). Since P (π(e)− J(e)) = 0, we must have π(e) = J(e).Therefore J = π is multiplicative and E = P is a normal conditional expectationfrom N onto π(M). �

Lp 2-ISOMETRIES 19

Lemma 4.4. Let N1 be a von Neumann subalgebra of N2 and let φ be a normalfaithful state on N2 with φ = φ|N1 . For 2 ≤ p ≤ ∞,∥∥∥φ 1

px∥∥∥Lp(N2)

≤∥∥∥φ 1

px∥∥∥Lp(N1)

for all x ∈ N1.

Proof. We first recall from Section 2 that we can identify Haagerup’s Lp-spaceLp(Ni) with the complex interpolation spaces [Ni, (Ni)∗] 1

p(i = 1, 2) (see (2.5)

and (2.6)). Since φ = φ|N1 and φ are normal faithful states on N1 and N2,respectively, the induced inclusion

ι2: φ12x ∈ L2(N1, φ) → φ

12x ∈ L2(N2, φ)

is an isometric inclusion. The corresponding inclusion

ι2: φx ∈ [N1, (N1)∗] 12→ φx ∈ [N2, (N2)∗] 1

2

between complex interpolation spaces is also an isometric inclusion. Then forany 2 ≤ p <∞ the canonical inclusion

ιp: φ1px ∈ Lp(N1) → φ

1px ∈ Lp(N2)

is a contraction since it can be identified with the complex interpolation

ιp = [ι∞, ι2] 2p,

where we let ι∞: N1 ↪→ N2 denote the canonical inclusion of N1 into N2. �

Getting back to T , let us assume that

T (φ1px) = φ

1pπ(x).

Otherwise, we may replace T with φ1px 7→ w∗T (φ

1px) where w is the partial

isometry obtain from the polar decomposition T (φ1p ) = wφ

1p . For any 1 ≤ q <∞,

we define the related maps

Tq(φ1q x) = φ

1q π(x) .

Corollary 4.5. Let 1 < p ≤ 2. If T = Tp: φ1px 7→ φ

1pπ(x) is an isometry, then

Tp′ is also an isometry.


Proof. Let us show that T′∗p′ Tp is the identity on Lp(M). Indeed, by definition

and Lemma 4.2 we find

trM(T′∗p′ (φ

1pπ(y))∗φ

1p′ x) = trM((φ

1pπ(y))∗Tp′(φ

1p′ x)) = trN (π(y∗)φ

1p φ

1p′ π(x))

= φ(π(xy∗)) = φ(xy∗) = trM((φ1p y)∗φ

1p′ x) .

By duality we conclude that T′∗p′ (φ

1pπ(y)) = φ

1p y. This implies that

T′∗p′ (Tp(φ

1p y))) = T

′∗p′ (φ

1pπ(x)) = φ

1p y.

This shows that T′∗p′ Tp = idLp(M). By duality, we deduce that

idLp′ (M) = (T′∗p′ Tp)

′∗ = T′∗p Tp′ .

By assumption Tp and thus T′∗p is a contraction. Tp′ is also contractive by Lemma

4.4, so it must be an isometry. �

Due to Corollary 4.5, we may now focus on the case 2 < p < ∞. The mainargument here is to show that if Tp is isometric for some value of p, then it isisometric for all values. We will use Kosaki’s interpolation theorem for the proofof this result but in a more explicit form. Let us use the notation S = {z ∈C| 0 ≤ Re(z) ≤ 1}.

Lemma 4.6. Let φ and ψ be normal faithful states such that ψ ≤ Cφ for someconstant C > 0. Let 2 < r < p < q <∞ and 1

p = 1−θq + θ

r . Then there exists ananalytic function h: S →M such that

(1)∥∥∥φ 1

q h(it)∥∥∥q≤ 1,

(2)∥∥∥φ 1

r h(1 + it)∥∥∥r≤ 1,

(3) φ1q h(0) = ψ

1q , φ

1ph(θ) = ψ

1p , φ

1r h(1) = ψ

1r ,

(4) the maps t 7→ h(1 + it)φ1r , t 7→ h(it)φ

1q are continuous.

Proof. Since ψ ≤ Cφ, we have by [44, Theorem VIII.3.17] (taking adjoints) thatthe Connes cocycle derivative (Dφ : Dψ)t = ψitφ−it extends off the real line toa σ-weakly continuous function on {z | 0 ≤ Im z ≤ 1/2} which is analytic in theinterior. We define 1

s = 1r −

1q and set

h(z) = (Dφ : Dψ)i( 1q + z

s ) = φ−1q−

zsψ

1q + z

s .

Note that h is analytic on S.

Lp 2-ISOMETRIES 21

We have ∥∥∥φ 1q h(it)

∥∥∥ =∥∥∥(φ− it

s ψits )ψ

1q

∥∥∥q

=∥∥∥ψ 1

q

∥∥∥q

= 1.

Similarly, ∥∥∥φ 1r h(1 + it)

∥∥∥r

=∥∥∥(φ− it

s ψits )ψ

1r

∥∥∥q

= 1 .

The equalities for φ1q h(0), φ

1r h(1) and φ

1ph(θ) are obvious. Since (φ−

its ψ

its ) is a

cocycle, it is strongly continuous. Then we obtain the last statement by Lemma2.1 and the above equations. �

We are now able to prove the key extrapolation result.

Proposition 4.7. Let φ be a normal faithful state on N , φ ◦ π = φ, andTp(φ

1px) = φ

1pπ(x). If Tp is an isometry for some 2 < p < ∞, then Tp is

an isometry for all 2 < p <∞.

Proof. Let 2 < r < p < q < ∞ and assume that Tp is an isometry. We want toshow that Tq and Tr are isometries. We define 1

s = 1r −

1q and θ by 1

p = 1−θq + θ

r .Let us assume that ψ is a normal faithful state with ψ ≤ Cφ for some constantC <∞. (Such ψ form a dense face of M+

∗ , see e.g. [13].) We know that Tq is acontraction by Lemma 4.4; assume toward a contradiction that∥∥∥Tq(ψ 1

q )∥∥∥q< 1 .

Since Tq is continuous, we can find a δ > 0 such that

(4.3)∥∥∥Tq(φ− it

s ψits ψ

1q )∥∥∥q≤ (1− δ)

for all |t| ≤ δ. Let µθ be the probability measure on the boundary of the strip S

such that ∫∂S

f(z)dµθ(z) = f(θ)

for every harmonic function on S. This is just a relocation of the Poisson ker-nel, so Lebesgue measure is absolute continuous with respect to µθ (an explicitformula is in [3, Section 4.3, p.93]). Therefore µθ([−iδ, iδ]) > 0. So we may alsofind ε > 0 such that 1−ε

1+ε > 1− δµθ[−iδ, iδ]. Since Lp′(N ) = [Lq′(N ), Lr′(N )]θ,

we may find an approximately norm-attaining element for Tp(ψ1p ) as a sim-

ple element of the interpolation space. That is, we find an analytic functiong : S → Lr′(N ), continuous on the boundary and vanishing at ∞ such that∥∥∥g(it)φ 1

r−1q

∥∥∥q′≤ (1 + ε), ‖g(1 + it)‖r′ ≤ (1 + ε),


and

1− ε ≤ |trN (g(θ)φ1r−

1pTp(ψ

1p ))| .

Let h be the function given by Lemma 4.6, and set

F (z) = trN (g(z)φ1r π(h(z))) = trN (g(z)Tr(φ

1r h(z))) .

Since z 7→ φ1r h(z) is analytic in the strip and continuous at the boundary and

bounded, we know that F is analytic, continuous at the boundary and vanishesat ∞. Therefore, we deduce that

1− ε ≤ |tr(g(θ)φ1r−

1pTp(φ

1ph(θ))| = |tr(g(θ)φ 1

r π(h(θ))| = |F (θ)|(4.4)

=∣∣∣∣∫ F (z)dµθ(z)

∣∣∣∣ ≤ ∫iR

|F (it)|dµθ(it) +∫

1+iR

|F (1 + it)|dµθ(it) .

For all t we have

|F (it)| = |trN (g(it)Tr(φ1r h(it))| = |trN (g(it)φ

1r π(h(it)))|

= |trN (g(it)φ1r−

1q Tq(φ

1q h(it))|

≤∥∥∥g(it)φ 1

r−1q

∥∥∥q′

∥∥∥φ 1q h(it)

∥∥∥q≤ (1 + ε) .

However, for |t| ≤ δ, we note by (4.3) that the inequality is stronger:

|F (it)| = |trN (g(it)Tr(φ1r h(it))| = |trN (g(it)φ

1r π(h(it)))|

= |trN (g(it)φ1r−

1q Tq(φ

1q h(it))|

≤∥∥∥g(it)φ 1

r−1q

∥∥∥q′

∥∥∥T (φ1q h(it))

∥∥∥q

≤ (1 + ε)∥∥∥T (φ−

its ψ

its ψ

1q )∥∥∥q

≤ (1 + ε)(1− δ) .

Similarly, we find

|F (1 + it)| ≤ 1 + ε .

By (4.4), this implies

1− ε ≤ (1 + ε)[(1− δ)µθ([−iδ, iδ]) + µθ(∂S \ [−iδ, iδ])]

= (1 + ε)[1− δµθ([−iδ, iδ])] ,

contradicting the choice of ε. This shows that∥∥∥Tq(ψ 1

q )∥∥∥q

= 1. The argument for∥∥∥Tr(ψ 1r )∥∥∥q

= 1 is similar.

Lp 2-ISOMETRIES 23

It was shown in [33] that the Mazur map ψ 7→ ψ1q is continuous. So the above

argument applies to a dense set of Lq(M)+. Since Tq is a contraction, we deducefor all positive elements ψ

1q∥∥∥Tq(ψ 1

q )∥∥∥Lq(N )

= ‖ψ1q ‖Lq(M) .

For an arbitrary element h ∈ Lq(M), we write hv∗ = |ξ∗| where v is a partialisometry. Then

‖h‖q = ‖h∗‖q = ‖ |h∗| ‖q = ‖Tq(|h∗|)‖q = ‖Tq(h)π(v∗)‖q ≤ ‖Tq(h)‖q .

Therefore Tq is an isometry. The same argument applies for Tr. �

Lemma 4.8. Keep the notations of the previous proposition, and let T4 be a2-isometry. Then for all x ∈M,∥∥∥φ 1

4π(x)φ14

∥∥∥2

=∥∥∥φ 1

4xφ14

∥∥∥2.

Proof. Let x be a positive element, then we know that∥∥∥φ 14π(x)φ

14

∥∥∥2

=∥∥∥φ 1

4π(x12 )∥∥∥2

4=∥∥∥φ 1

4x12

∥∥∥2

4=∥∥∥φ 1

4xφ14

∥∥∥2.

Given an arbitrary element x ∈M of norm less than one, the matrix

x =[

1 xx∗ 1

]is positive. We apply the observation above to x and tr2⊗φ as a normal faithfulstate on M2(M) and deduce∥∥∥(tr2 ⊗ φ)

14π2(x)(tr2 ⊗ φ)

14

∥∥∥2

=∥∥∥(tr2 ⊗ φ)

14 x(tr2 ⊗ φ)

14

∥∥∥2.

Since∥∥∥φ 1

4x∗φ14

∥∥∥2

2=∥∥∥φ 1

4xφ14

∥∥∥2

2, we have∥∥∥(tr2 ⊗ φ)

14 x(tr2 ⊗ φ)

14

∥∥∥2

2

= (tr2 ⊗ trM)([φ

14 0

0 φ14

] [1 x∗

x 1

] [φ

12 0

0 φ12

] [1 xx∗ 1

] [φ

14 0

0 φ14

])= (tr2 ⊗ trM)

([φ+ φ

14x∗φ

12xφ

14 φ

34xφ

14 + φ

14x∗φ

34

φ34xφ

14 + φ

14x∗φ

34 φ+ φ

14xφ

12x∗φ

14

])=

12

(2trM(φ) + trM(φ

14x∗φ

12xφ

14 ) + trM(φ

14xφ

12x∗φ

14 ))

= 1 +∥∥∥φ 1

4xφ14

∥∥∥2

2.


Since φ(π(1)) = 1, the same calculation shows that∥∥∥(tr2 ⊗ φ)14π2(x)(tr2 ⊗ φ)

14

∥∥∥2

2=(

1 +∥∥∥φ 1

4π(x)φ14

∥∥∥2

2

).

This proves the assertion. �

At this point we recall how complete isometries of Lp-spaces can be constructedfrom either *-isomorphisms or conditional expectations. Details for the construc-tions of the next two paragraphs can be found in [11], [16], and [38].

Let π: M1 →M2 be a surjective *-isomorphism, and let φ ∈ (M1)+∗ be faith-ful. Then there is an associated completely isometric isomorphism πp: Lp(M1)

∼→Lp(M2), densely defined by

φ1px 7→ (φ ◦ π−1)

1pπ(x), x ∈M1.

Now consider the situation where M1 is a conditioned σ-finite subalgebra ofM2, so that there are a normal *-isomorphism ι: M1 ↪→ M2 (thought of asthe identity map) and a normal conditional expectation E: M2 � M1. Byinterpolation one can find a complete isometry ιp: Lp(M1) ↪→ Lp(M2) and acomplete contraction Ep: Lp(M2) � ιp(Lp(M1)). Both ιp and Ep are completelypositive. With φ ∈ (M1)+∗ faithful, ιp is densely defined by

φ1px 7→ (φ ◦ E)

1px, x ∈M1.

Theorem 4.9. Let 1 ≤ p 6= 2 < ∞. A linear map T : Lp(M) → Lp(N ) is a2-isometry if and only if there exist a normal ∗-monomorphism π: M → N , apartial isometry w ∈ N , and a normal conditional expectation E: N → π(M)such that π(1) = w∗w and

(4.5) T (φ1px) = w(φ ◦ π−1 ◦ E)

1pπ(x), ∀x ∈M.

Under these conditions, the range T (Lp(M)) is completely contractively comple-mented, and T is a module map:

T (hx) = T (h)π(x), ∀x ∈M, h ∈ Lp(M).

Proof. First note that maps of the form (4.5) are always complete isometrieswith completely contractively complemented range: they decompose as πp, thenιp, then left multiplication by w. Each of these is a complete isometry, and theimage is the range of the complete contraction Lp(M2) 3 h 7→ wEp(w∗h).

Lp 2-ISOMETRIES 25

Now we turn to the derivation of (4.5) for 2-isometries. The case p = 1 hasbeen proved in Proposition 4.3. To prove the result for general p, we may firstapply Proposition 4.1 and assume that

T (φ1px) = φ

1pπ(x) .

We can obtain the result for general T by multiplying by an appropriate partialisometry.

According to Lemma 4.2, we also know that φ = φ ◦ π. If T is a 2-isometryfor some 1 < p 6= 2 <∞, then we can claim from Corollary 4.5 and Proposition4.7 that T4 is a 2-isometry. Then Lemma 4.8 shows that∥∥∥φ 1

4xφ14

∥∥∥2

=∥∥∥φ 1

4π(x)φ14

∥∥∥2.

By polarization, we deduce

(φ14xφ

14 , φ

14 yφ

14 ) = (φ

14π(x)φ

14 , φ

14π(y)φ

14 )

for all x, y ∈M. This means that

trM(φ12xφ

12 y∗) = trN (φ

12π(x)φ

12π(y)∗) .

In other words the sesquilinear selfpolar form [51] sφ(x, y) , tr(φ12xφ

12 y∗) satis-

fies

(4.6) sφ|π(M)×π(M) = sφ|π(M).

Now by a result of Haagerup and Størmer [10, Theorem 4.2], equation (4.6) andthe equality π(1) = s(φ) imply the existence of a faithful normal conditionalexpectation F : π(1)Nπ(1) → π(M) such that φ = φ ◦ F . Defining E: x 7→F (π(1)xπ(1)), x ∈ N , we have

φ = φ ◦ E = φ ◦ π−1 ◦ E,

which establishes (4.5). �

Proof of Theorem 2. The main difference between Theorem 2 and Theorem 4.9is that T is described on the spanning set of positive vectors {ϕ

1p | ϕ ∈ M+

∗ }instead of the dense set {φ

1px | x ∈ M} (and T becomes everywhere-defined by

linearity). The equation (1.2) is a noncommutative version of (1.1), so that Tmay be naturally viewed as a “noncommutative weighted composition operator”.To establish (1.2), it is sufficient to show that the data π,E,w of Theorem 4.9do not depend on the choice of φ ∈M+

∗ .


For this, choose an arbitrary faithful ψ ∈ M+∗ satisfying ψ

2p ≤ Cφ

2p for some

C <∞. (Again, this set is dense.) The assumption means that d , (Dφ : Dψ)i/pexists in M, and by analytic continuation we have φ

1p d = ψ

1p . (In fact the

equation d = φ−1pψ

1p is justified rigorously in [40].) Cocycles are functorial

with respect to normal *-isomorphisms, and they are invariant when weights areprecomposed with a normal conditional expectation [44, Corollary IX.4.22], so

π(d) = (D(φ ◦ π−1 ◦ E) : D(ψ ◦ π−1 ◦ E))i/p.

Then we have

T (ψ1p ) = T (φ

1p d) = w(φ ◦ π−1 ◦ E)

1pπ(d) = w(ψ ◦ π−1 ◦ E)

1p .

By density, this establishes (1.2). See also [38, Section 6].Finally we note that the formulation of Theorem 2 does not require the σ-

finiteness of M. For each q in the net of σ-finite projections, the restricted isom-etry T : Lp(qMq) = qLp(M)q → Lp(N ) is of the form (1.2) for some wq, πq, Eq.One only needs to glue them all together. This can be done in exactly the sameway as in the proof of Theorem 3.1. �

5. Concluding remarks

Remark 1. If, in the setup of Theorem 2, we require that w∗w = π(1) is thesupport of E, then π, w, and E are uniquely determined. For suppose thatπ1, E1, w1 also define the 2-isometry T , and assume that s(E1) = π1(1). Thenfor all ϕ ∈M+

∗ ,

w(ϕ ◦ π−1 ◦ E)1p = w1(ϕ ◦ π−1

1 ◦ E1)1p ⇒ ϕ ◦ π−1 ◦ E = ϕ ◦ π−1

1 ◦ E1

⇒ π−1 ◦ E = π−11 ◦ E1 ⇒ id = π−1 ◦ E ◦ π1 ⇒ π = E ◦ π1.

Now take a projection p ∈M and calculate

E(π(p)π1(p)π(p)) = π(p)E(π1(p))π(p) = π(p)3 = π(p) = E(π(p)).

Since E is faithful on π(1)Nπ(1), we must have π(p)π1(p)π(p) = π(p), or π1(p) ≥π(p). Reversing the argument proves the equality of π and π1, and the other datamust be equal also (subject to w∗w = π(1) = w∗1w1).

Remark 2. Given such a 2-isometry T decomposed as in Theorem 2, the asso-ciated map

S: Lp(M) → Lp(N ), h 7→ w∗T (h),

Lp 2-ISOMETRIES 27

is a completely positive complete isometry whose range is completely positivelyand completely contractively complemented. If T is positive, then w = π(1) andT = S (and in particular T is already completely positive).

Remark 3. The arguments in Section 4 can be used to establish the followingresult, which seems to be of independent interest. The main step strengthensLemma 4.8 by only requiring T to be an isometry (i.e. not a 2-isometry).

Proposition 5.1. Let M ⊂ N be a unital inclusion of von Neumann algebraswith φ ∈M+

∗ , φ ∈ N+∗ , both faithful, satisfying φ |M= φ. Assume that the map

Tp: Lp(M) → Lp(N ); φ1px 7→ φ

1px, x ∈M,

is isometric for some 1 ≤ p 6= 2 < ∞. Then there exists a faithful normalconditional expectation E: N →M such that φ = φ ◦ E.

Proof. It follows from Lemma 4.4, Corollary 4.5, and Proposition 4.7 that if Tpis isometric for one p in the given range, it is isometric for all. In particular, T4

is isometric, so that for any M3 y ≥ 0,

(5.1) ‖φ 14 yφ

14 ‖ = ‖φ 1

4 y12 ‖2 = ‖φ 1

4 y12 ‖2 = ‖φ 1

4 yφ14 ‖.

Suppose that y ∈M is only self-adjoint. For all scalar t ≥ ‖y‖, (5.1) implies

‖φ 14 yφ

14 ‖2 + 2tφ(y) + t2‖φ‖ = ‖φ 1

4 (y + t1)φ14 ‖2

= ‖φ 14 (y + t1)φ

14 ‖2

= ‖φ 14 yφ

14 ‖2 + 2tφ(y) + t2‖φ‖.

Since these are polynomials in t which agree on a half-line, they must have thesame constant terms. We conclude that (5.1) holds for all self-adjoint y ∈M.

In other words the map φ14 yφ

14 7→ φ

14 yφ

14 (with y ∈ Msa) is isometric be-

tween the real Hilbert spaces L2(M)sa and L2(N )sa. It therefore preserves innerproducts, which is the same as saying that the self-polar form sφ agrees with sφrestricted to M×M. Then the Haagerup-Størmer result [10, Theorem 4.2] againimplies the existence of a faithful normal conditional expectation E: N → Msatisfying φ = φ ◦ E. �

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Department of Mathematics, University of Illinois, Urbana, IL 61801, USAE-mail address, Marius Junge: [email protected]

E-mail address, Zhong-Jin Ruan: [email protected]

E-mail address, David Sherman: [email protected]

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